How Many Bananas Does It Take To Lead A Camel To Market?

Welcome to The Riddler. Every week, I offer up a problem related to the things we hold dear around here: math, logic and probability. These problems, puzzles and riddles come from many top-notch puzzle folks around the world — including you! You’ll find this week’s puzzle below.

Mull it over on your commute, dissect it on your lunch break and argue about it with your friends and lovers. When you’re ready, submit your answer using the link below. I’ll reveal the solution next week, and a correct submission (chosen at random) will earn a shoutout in this column. Important small print: To be eligible, I need to receive your correct answer before 11:59 p.m. EDT on Sunday. Have a great weekend!

Before we get to the new puzzle, let’s return to last week’s. Congratulations to 👏 Christopher Tiee 👏 of San Diego, our big winner. You can find a solution to the previous Riddler at the bottom of this post.

Now here’s this week’s Riddler, a twist on a classic passed along by Scott Rodilitz, a Ph.D. student in operations at Yale.

You have a camel and 3,000 bananas. (Because of course you do.) You would like to sell your bananas at the bazaar 1,000 miles away. Your loyal camel can carry at most 1,000 bananas at a time. However, it has an insatiable appetite and quite the nose for bananas — if you have bananas with you, it will demand one banana per mile traveled. In the absence of bananas on his back, it will happily walk as far as needed to get more bananas, loyal beast that it is. What should you do to get the largest number of bananas to the bazaar? What is that number?

Extra credit: Let’s push this classic even further and offer up a 🏆 Coolest Riddler Extension Award 🏆. Add a second camel, another fruit, a closer but less profitable bazaar, or something even more creative. Submit your extension and its solution via the form below. The winner gets a shiny emoji trophy next week.

Submit your answer
Need a hint? You can try asking me nicely. Want to submit a new puzzle or problem? Email me. I’m especially on the hunt for Riddler Express problems — bite-size puzzles that don’t take quite as much time or computational power to solve.

And here’s the solution to last week’s Riddler, concerning a very particular mathematician, his birthday cake and the memento he wants to fit the cake in. If you make the largest three-tier cake that fits under the mathematician’s glass cone, it will fill about 70.2 percent of the cone’s volume. If the cone is 1 unit tall, the heights of the tiers, from bottom to top, should be roughly 0.162, 0.182 and 0.219.

This is a constrained optimization problem. We want to optimize the volume of the cake, because a mathematician told us to and cake is delicious, subject to the constraints of the dimensions of the cone.

Here’s how to do that, adapted from the solution of the puzzle’s submitter, Jim Crimmins. Let \(A_B\) be the area of the cone’s base and \(H\) be its height. The volume of the cone is therefore \(V_C=(1/3)A_B H\).

Let the heights of the cake layers be \(a\), \(b\) and \(c\), expressed as percentages of \(H\). The volumes of the three layers are then:

$$V_a = ((1-a)^2 A_B) \cdot aH$$

$$V_b = ((1-a-b)^2 A_B) \cdot bH$$

$$V_c = ((1-a-b-c)^2 A_B) \cdot cH$$

The total volume of the cake, \(V_T\), is the sum of those three. The percentage of the cone the cake fills — the number we want to maximize — is \(P\).

$$P=V_T/V_C=3\left(a(1-a)^2+b(1-a-b)^2+c(1-a-b-c)^2\right)$$

We want to maximize \(P\) subject to \(a+b+c\leq 1\). (The total height of the layers can’t exceed the height of the cone, after all.) Solving that — basically un-repressing your memories of calculus class and taking partial derivatives and setting them equal to zero — gives us our optimal heights and our optimal volume.

For extra credit, I asked what the cake would look like if it had N layers. The proportion of the cone we can fill as N goes to infinity is clearly 1. The precise math on the way to infinity gets pretty messy, and I’m not aware of a closed-form solution for the optima. However, you can see these nice approaches from Joachim Worthington and Daniel Filan for two examples out of the many nice ones I received. And here is a chart of the optimized volumes, as the number of cake tiers increases, from Laurent Lessard: